How could I find the intersection between a face and an edge?
Not sure if this is what you mean, but the TinyCAD extension might be able to do this.

It doesn't have an "intersect face with edge" operator, but as a workaround it can extend an edge to a face, so you might be able to do this by shortening your edge enough that it no longer intersects said face, then extend it.

Snap the 3D cursor at the end of said edge, set pivot to 3D cursor and scale the opposite vertex towards it.

You can then press W Specials > TinyCAD > Extend Edge to Face